\(\mathrm {K}\) -theory is rooted in the classification of vector bundles over a topological space \(X\)  up to homotopy. Assuming that \(X\)  is compact and Hausdorff, these can be thought of as homotopy classes of projections in \(\mathrm {M}_n(\mathrm {C}(X))\)  for large enough \(n\) . Positive elements in this algebra with the same Cuntz class must at least have the same rank at every point of \(X\) , and so homotopies as in Theorem REF  would have to preserve rank pointwise. Such homotopies are obstructed by the \(\mathrm {K}\) -theory of closed subsets of \(X\) . Indeed, this is the fundamental observation of [1]}. Our result says that for positive elements as in Theorem REF  which are not homotopic to projections, this obstruction vanishes—these elements behave like trivial vector bundles, with their homotopy class determined only by their pointwise rank with respect to lower semicontinuous dimension functions.
We expect that the conditions of real rank zero and approximate divisibility can be relaxed to the property of \(\mathcal {Z}\) -stability.
